Page 162 - Fiber Bragg Gratings

4.2 Coupled-mode theory 139

We can estimate the maximum angle for the radiation by observing
the point E on the phase-matching curve in Fig. 4.6. The tangent to this
point on the phase matching at E intersects the cladding circle at the
"+" point. This point subtends the largest radiation mode angle for this
particular grating, at the origin. The maximum angle of the radiation for
an untilted grating is at the shortest wavelength and is easily shown to
be

which is maximum if n eff = n clad. For a core-to-cladding refractive index
difference of 0.01 in a silica fiber, ff max *» 6.7°. It should be remembered
that phase matching to specific radiation modes will only occur if a clad-
ding mode exists with the appropriate mode index. However, with an
infinite cladding, coupling to a continuum of the radiation field occurs so
that the spectrum is continuous.
There is another possibility for coupling to radiation modes. We begin
with 6 g = 0 and the condition for Bragg reflection from, for example, the
forward to the counterpropagating LP Ql mode. If the grating is tilted at
m
an angle 6 g, it is shown simply as a rotation of n g around the pivot at G .
Following the mathematical approach taken for Eq. (4.2.33), we find that
at some angle ff g the radiation mode is at the Bragg wavelength, i.e., the
start wavelength moves toward the Bragg wavelength, until they coincide.
At this point, there is strong coupling to the radiation modes. Referring
to Fig. 4.6, the angle is easily found by changing the tilt of the grating.
This directly leads to

where n eff is the effective index of the mode at the Bragg wavelength of
the untilted grating (when O g = 0), so that

Again, the tilt angle of the grating for this condition to be met in-
creases with n eff and is a maximum when n eff = n core. We can calculate